Poisson Distribution Calculator
The probability of exactly k events, at most k, fewer than k, at least k or more than k, when events happen at an average rate λ per interval.
Result
P(X = k), exactly k: 22.4042%
Assumes events are independent and arrive at a steady average rate. Clustered or seasonal events are not Poisson.
- P(X ≤ k), at most k
- 42.319%
- P(X < k), fewer than k
- 19.9148%
- P(X ≥ k), at least k
- 80.0852%
- P(X > k), more than k
- 57.681%
- Mean, λ
- 3
- Standard deviation, √λ
- 1.732051
https://calcgallery.com/statistics/poisson-distribution-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Poisson Distribution Calculator
- Enter λ, the average number of events in the interval you are asking about.
- Enter k, the number of events whose chance you want.
- Read the chance of exactly k, and of at most, fewer than, at least and more than k.
The formula
P(X = k) = λ^k × e^−λ ÷ k!P(X ≤ k) = P(X = 0) + P(X = 1) + … + P(X = k)P(X ≥ k) = P(X = k) + P(X = k + 1) + …mean = λ standard deviation = √λThe mean and the variance of a Poisson count are both λ, so the standard deviation is √λ. With λ = 100 a count is usually within about 10 of 100 either way.
Large and small numbers
Each term is worked out with logarithms, so λ and k in the thousands do not overflow. The upper tail is added up directly rather than taken as one minus the lower tail, which would lose a probability of one in a billion to rounding. Probabilities are shown as percentages to four decimal places. When that rounds a chance far out in the tail to 0%, or a near certainty to 100%, the exact figure is given below it in scientific notation: at λ = 4, at least 20 events is 1.02 × 10^-6%, not 0%.
What the diagram shows
A bar for each possible count, its height the probability of that count, with the bar for k picked out. When λ is large, only a window of 40 bars around λ and k is drawn.
With your numbers
- P(X = k) = λ^k × e^−λ ÷ k! = 3^2 × e^−3 ÷ 2! = 0.224042
- P(X ≤ k) = P(X = 0) + … + P(X = 2) = 0.42319
Quick reference
Assumes events are independent and arrive at a steady average rate. Clustered or seasonal events are not Poisson.
| Example | P(X = k), exactly k |
|---|---|
| λ = 3, exactly 2 | 22.4042% |
| λ = 2.5, none | 8.2085% |
| λ = 10, 15 events | 3.4718% |
| λ = 0.5, 3 events | 1.2636% |
| λ = 100, 100 events | 3.9861% |
| λ = 4, 20 events | 0% |
| λ = 1,000, 1,000 events | 1.2615% |
Common questions about the Poisson Distribution Calculator
- When does the Poisson distribution apply?
- When you are counting events that happen one at a time, independently of each other, at a steady average rate: calls to a help line in an hour, typos per page, arrivals at a counter. It does not fit events that cluster, such as orders after an advert, or a rate that changes through the interval.
- How do I get λ for a different interval?
- Scale the rate to the interval. If a counter averages 2 arrivals a minute and you ask about 5 minutes, λ is 10. The Poisson mean and the interval always go together, so a rate per hour cannot be used for a question about a day without multiplying it up first.
- What is the difference between at most and fewer than?
- At most k includes k itself; fewer than k does not. With λ = 3, at most 2 events has a probability of about 42.3%, and fewer than 2 about 19.9%. The two differ by exactly the chance of 2 events, about 22.4%.
- How does Poisson relate to the binomial?
- The binomial counts successes in a fixed number of trials. When the trials are many and each success is rare, the binomial with n trials and chance p is close to Poisson with λ = n × p. The probability calculator on this site works out exact binomial chances.